There are two principal directions that model geometrical structures underlying the manifold on which the discrete probability measures lie [1]}. The first direction is based on the principal of invariance, which relies on the criterion that the geometry between probability measures should be invariant under invertible transformations of random variables. This perspective is the foundation of the theory of information geometry, which operates as a base for the statistical inference [2]}. The second direction is established by the theory of Optimal Transport (OT), which exploits prior geometric knowledge on the base space in which random variables are valued [3]}. Computing OT or Wasserstein distance between two random variables equals to achieving a coupling between these two variables that is optimal in the sense that the expectation of the transportation cost between the first and second variables is minimal. The Wasserstein distance between two probability measures considers the metric properties of the base space on which a structure or a pattern is defined. However, traditional information-theoretic divergences such as the Hellinger divergence and the Kullback-Leibler (KL) divergence are not able to properly capture the geometry of the base space. Thus, the Wasserstein distance is useful for the applications where the structure or geometry of the base space plays a significant role [4]}. In this work, similar to other SSL methods, we make a structural assumption about the data in which the data are represented by a CNN model. Inspired by the Wasserstein distance, which exploits properly the geometry of the base space to provide a natural notion of similarity between the discrete empirical measures, we use it to provide pseudo-labels for the unlabeled data to train a CNN model in an SSL fashion. Specifically, in our SSL method, labeled data belonging to each class is a discrete measure. Thus, all the labeled data create a measure of measures and similarly, the pool of unlabeled data is also a measure of measures constructed by data belonging to different classes. Thus, we design a measure of measures OT plan serving as a similarity metric between discrete empirical measures to map the unlabeled measures to the labeled measures based on which, the pseudo-labels for the unlabeled data are inferred. Our SSL method is based on the role of Wasserstein distances in the hierarchical modeling [5]}.
It stems from the fact that the labeled and unlabeled datasets hierarchically create a measure of measures in which each measure is constructed by the data belonging to the same class.
